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15
Compact Routing in PowerLaw Graphs
"... Abstract. We adapt the compact routing scheme by Thorup and Zwick to optimize it for powerlaw graphs. We analyze our adapted routing scheme based on the theory of unweighted random powerlaw graphs with fixed expected degree sequence by Aiello, Chung, and Lu. Our result is the first theoretical bou ..."
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Cited by 15 (3 self)
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Abstract. We adapt the compact routing scheme by Thorup and Zwick to optimize it for powerlaw graphs. We analyze our adapted routing scheme based on the theory of unweighted random powerlaw graphs with fixed expected degree sequence by Aiello, Chung, and Lu. Our result is the first theoretical bound coupled to the parameter of the powerlaw graph model for a compact routing scheme. In particular, we prove that, for stretch 3, instead of routing tables with Õ(n 1/2) bits as in the general scheme by Thorup and Zwick, expected sizes of O(n γ log n) bits are sufficient, and that all the routing tables can be constructed at once in expected time O(n 1+γ log n), with γ = τ−2 + ε, where τ ∈ (2, 3) 2τ−3 is the powerlaw exponent and ε> 0. Both bounds also hold with probability at least 1 − 1/n (independent of ε). The routing scheme is a labeled scheme, requiring a stretch5 handshaking step and using addresses and message headers with O(log n log log n) bits, with probability at least 1−o(1). We further demonstrate the effectiveness of our scheme by simulations on realworld graphs as well as synthetic powerlaw graphs. With the same techniques as for the compact routing scheme, we also adapt the approximate distance oracle by Thorup and Zwick for stretch 3 and obtain a new upper bound of expected Õ(n1+γ) for space and preprocessing. 1
Optimal scalefree compact routing schemes in doubling networks
 In ACMSIAM symposium on Discrete algorithms
, 2007
"... We consider compact routing schemes in networks of low doubling dimension, where the doubling dimension is the least value α such that any ball in the network can be covered by at most 2 α balls of half radius. There are two variants of routing scheme design: (i) labeled (namedependent) routing, wh ..."
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Cited by 9 (2 self)
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We consider compact routing schemes in networks of low doubling dimension, where the doubling dimension is the least value α such that any ball in the network can be covered by at most 2 α balls of half radius. There are two variants of routing scheme design: (i) labeled (namedependent) routing, where the designer is allowed to rename the nodes so that the names (labels) can contain additional routing information, e.g. topological information; and (ii) nameindependent routing, which works on top of the arbitrary original node names in the network, i.e. the node names are independent of the routing scheme. In this paper, given any constant ǫ ∈ (0, 1), and an nnode weighted network of low doubling dimension α ∈ O(loglog n), we present • A (1+ǫ)stretch labeled compact routing scheme with ⌈log n⌉bit routing labels, O(log 2 � n/log log n)bit packet headers, andbit routing information at each node; ( 1 ǫ)O(α) log 3 n • A (9 + ǫ)stretch nameindependent compact routing scheme with O(log 2 � n/log log n)bit packet headers, andbit routing information at each node. ( 1 ǫ)O(α) log 3 n In addition, we also prove a lower bound: any nameindependent routing scheme with o(n (ǫ/60)2) bits of storage at each node has stretch no less than 9 −ǫ, for any ǫ ∈ (0, 8). Therefore our nameindependent routing scheme achieves asymptotically optimal stretch with polylogarithmic storage at each node and packet headers. Note that both schemes are scalefree in the sense that their space requirements do not depend on the normalized diameter ∆ of the network. We also present a simpler nonscalefree (9 + ǫ)stretch nameindependent compact routing scheme with improved space requirements if ∆ is polynomial in n. 1
The Inframetric Model for the Internet
, 2007
"... Abstract—A large amount of algorithms has recently been designed for the Internet under the assumption that the distance defined by the roundtrip delay (RTT) is a metric. Moreover, many of these algorithms (e.g., overlay network construction, routing scheme design, sparse spanner construction) rely ..."
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Cited by 8 (3 self)
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Abstract—A large amount of algorithms has recently been designed for the Internet under the assumption that the distance defined by the roundtrip delay (RTT) is a metric. Moreover, many of these algorithms (e.g., overlay network construction, routing scheme design, sparse spanner construction) rely on the assumption that the metric has bounded ball growth or bounded doubling dimension. This paper analyzes the validity of these assumptions and proposes a tractable model matching experimental observations. On the one hand, based on Skitter data collected by CAIDA and King matrices of Meridian and P2PSim projects, we verify that the ball growth of the Internet, as well as its doubling dimension, can actually be quite large. Nevertheless, we observed that the doubling dimension is much smaller when restricting the measures to balls of large enough radius. Moreover, by computing the number of balls of radius r required to cover balls of radius R> r, we observed that this number grows with R much slower than what is predicted by a large doubling dimension. On the other hand, based on data collected on the PlanetLab platform by the AllSitesPings project, we confirm that the triangle inequality does not hold for a significant fraction of the nodes. Nevertheless, we demonstrate that RTT measures satisfy a weak version of the triangle inequality: there exists a small constant ρ such that for any triple u, v, w, we have RTT(u,v) ≤ ρ ·max{RTT(u,w), RTT(w,v)}. (Smaller bounds on ρ can even be obtained when the triple u, v, w is skewed). We call inframetric a distance function satisfying this latter inequality. Inframetrics subsume standard metrics and ultrametrics. Based on inframetrics and on our observations concerning the doubling dimension, we propose an analytical model for Internet RTT latencies. This model is tuned by a small set of parameters concerning the violation of the triangle inequality and the geometrical dimension of the network. We demonstrate the tractability of our model by designing a simple and efficient compact routing scheme with low stretch. Precisely, the scheme has constant multiplicative stretch and logarithmic additive stretch. I.
Tradeoffs between Stretch Factor and Load Balancing Ratio in Routing on Growth Restricted Graphs
, 2004
"... A graph has growth rate k if the number of nodes in any subgraph with diameter r is bounded by O(r k). The communication graphs of wireless networks and peertopeer networks often have small growth rate. In this paper we study the tradeoff between two quality measures for routing in growth restrict ..."
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Cited by 7 (1 self)
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A graph has growth rate k if the number of nodes in any subgraph with diameter r is bounded by O(r k). The communication graphs of wireless networks and peertopeer networks often have small growth rate. In this paper we study the tradeoff between two quality measures for routing in growth restricted graphs. The two measures we consider are the stretch factor, which measures the lengths of the routing paths, and the load balancing ratio, which measures how evenly the traffic is distributed. We show that if the routing algorithm is required to use paths with stretch factor c, then its load balancing ratio is bounded by O((n/c) 1−1/k), where k is the graph’s growth rate. We illustrate our results by focusing on the unit disk graph for modeling wireless networks in which two nodes have direct communication if their distance is under certain threshold. We show that if the maximum density of the nodes is bounded by ρ, there exists routing scheme such that the stretch factor of routing paths is at most c, and the maximum load on the nodes is at most O(min ( � ρn/c, n/c)) times the optimum. In addition, the bound on the load balancing ratio is tight in the worst case. As a special case, when the density is bounded by a constant, the shortest path routing has a load balancing ratio of O (√n). The result extends to kdimensional unit ball graphs and graphs with growth rate k. We also discuss algorithmic issues for load balanced short path routing and for load balanced routing in spanner graphs.
Routing, Anycast, and Multicast for Mesh and Sensor Networks
"... Abstract — This paper studies routing schemes and their distributed construction in limited wireless networks, such as sensor or mesh networks. We argue that the connectivity of such networks is well captured by a constant doubling metric and present a constant stretch multicast algorithm through wh ..."
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Cited by 2 (0 self)
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Abstract — This paper studies routing schemes and their distributed construction in limited wireless networks, such as sensor or mesh networks. We argue that the connectivity of such networks is well captured by a constant doubling metric and present a constant stretch multicast algorithm through which any network node u can send messages to an arbitrary receiver set U. In other words, we describe a distributed approximation algorithm which is only a constant factor off the NPhard Minimum Steiner Tree on u ∪U. As a building block for the multicasting, we construct a 1 + ε stretch labeled routing scheme with label size O(logΘ) and storage overhead O(1/ε) α (logΘ)(O(α) + log∆), where Θ is the diameter of the network, ∆ the maximum degree of any network node, and α a constant representing the doubling dimension of the network. In addition to unicast and multicast, we present a constant approximation for anycasting on the basis of √ 6approximate distance queries. We provide a distributed algorithm to construct the required labeling and routing tables. I.
Spatial Distribution in Routing Table Design for Sensor Networks
"... Abstract—We propose a generic routing table design principle for scalable routing on networks with bounded geometric growth. Given an inaccurate distance oracle that estimates the graph distance of any two nodes with constant factor upper and lower bounds, we augment it by storing the routing paths ..."
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Cited by 2 (2 self)
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Abstract—We propose a generic routing table design principle for scalable routing on networks with bounded geometric growth. Given an inaccurate distance oracle that estimates the graph distance of any two nodes with constant factor upper and lower bounds, we augment it by storing the routing paths of pairs of nodes, selected in a spatial distribution, and show that the routing table enables 1 + ε stretch routing. In the wireless ad hoc and sensor network scenario, the geographic locations of the nodes serve as such an inaccurate distance oracle. Each node p selects O(log n loglog n) other nodes from a distribution proportional to 1/r 2 where r is the distance to p and the routing paths to these nodes are stored on the nodes along these paths in the network. The routing algorithm selects links conforming to a set of sufficient conditions and guarantees with high probability 1+ε stretch routing with routing table size O ( √ n log n loglog n) on average for each node. This scheme is favorable for its simplicity, generality and blindness to any global state. It is a good example that global routing properties emerge from purely distributed and uncoordinated routing table design. I.
The emergence of sparse spanners and greedy wellseparated pair decomposition. The 18th Fall Workshop on
 Combinatorial Rigidity. Graduate Studies in Math., AMS
, 1993
"... A spanner graph on a set of points in Rd contains a shortest path between any pair of points with length at most a constant factor of their Euclidean distance. A spanner with a sparse set of edges is thus a good candidate for network backbones, as desired in many practical scenarios such as the tran ..."
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Cited by 2 (2 self)
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A spanner graph on a set of points in Rd contains a shortest path between any pair of points with length at most a constant factor of their Euclidean distance. A spanner with a sparse set of edges is thus a good candidate for network backbones, as desired in many practical scenarios such as the transportation network and peertopeer network overlays. In this paper we investigate new models and aim to interpret why good spanners ‘emerge ’ in reality, when they are clearly built in pieces by agents with their own interests and the construction is not coordinated. Our main result is to show that the following algorithm generates a (1 + ε)spanner with a linear number of edges, constant average degree, and the total edge length as a small logarithmic factor of the cost of the minimum spanning tree. In our algorithm, the points build edges at an arbitrary order. When a point p checks on whether the edge to a point q should be built, it will build this edge only if there is no existing edge p ′ q ′ with p ′ and q ′ 1 at distances no more than 4(1+1/ε) · p ′ q ′ from p, q respectively. Eventually when all points have finished checking edges to all other points, the resulted collection of edges forms a sparse spanner as desired. This new spanner construction algorithm can be extended to a metric space with constant doubling dimension and admits a local routing scheme to find the short paths. As a side product, we show a greedy algorithm for constructing linearsize wellseparated pair decompositions that may be of interest on its own. A wellseparated pair decomposition is a collection of subset pairs such that each pair of point sets is fairly far away from each other compared with their diameters and that every pair of points is ‘covered ’ by at least one wellseparated pair. Our greedy algorithm selects an arbitrary pair of points that have not yet been covered and puts a ‘dumbbell ’ around the pair as the wellseparated pair, repeats this until all pairs of points are covered. When the algorithm finishes, we show only a linear number of pairs is generated, which is asymptotically optimal. 1
UltraLowDimensional Embeddings for Doubling Metrics
"... We consider the problem of embedding a metric into lowdimensional Euclidean space. The classical theorems of Bourgain and of Johnson and Lindenstrauss imply that any metric on n points embeds into an O(log n)dimensional Euclidean space with O(log n) distortion. Moreover, a simple “volume ” argumen ..."
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Cited by 2 (0 self)
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We consider the problem of embedding a metric into lowdimensional Euclidean space. The classical theorems of Bourgain and of Johnson and Lindenstrauss imply that any metric on n points embeds into an O(log n)dimensional Euclidean space with O(log n) distortion. Moreover, a simple “volume ” argument shows that this bound is nearly tight: the uniform metric on n points requires Ω(log n / log log n) dimensions to embed with logarithmic distortion. It is natural to ask whether such a volume restriction is the only hurdle to lowdimensional lowdistortion embeddings. Do doubling metrics, which do not have large uniform submetrics, embed in low dimensional Euclidean spaces with small distortion? In this paper, we answer the question positively and show that any doubling metric embeds into O(log log n) dimensions with o(log n) distortion. In fact, we give a suite of embeddings with a smooth tradeoff between distortion and dimension: given an npoint metric (V, d) with doubling dimension dimD, and any target dimension T in the range Ω(dimD log log n) ≤ T ≤ O(log n), we embed the metric into Euclidean space R T with O(log n � dimD /T) distortion.
Communication Network]: Distributed Systems General Terms: Algorithms, Design
"... A routing scheme on the graph G = (V, E) is a distributed algorithm that allows any source node to send packets to any destination node along the links of E. A routing scheme on a metric space (M, d) builds a graph G = (M, E) by distributedly selecting the edges (u, v) to be in E and routes only alo ..."
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A routing scheme on the graph G = (V, E) is a distributed algorithm that allows any source node to send packets to any destination node along the links of E. A routing scheme on a metric space (M, d) builds a graph G = (M, E) by distributedly selecting the edges (u, v) to be in E and routes only along the edges of E. The stretch of a routing path is its length divided by the length of the shortest path between its endpoints. The stretch of a routing scheme is the maximum stretch of a routing path. A routing scheme is compact if the routing table and packet header size are both polylog(M). Compact routing research has recently focused on graphs of low doubling dimension [10, 3, 8, 2, 5, 6, 7, 9], (the doubling dimension of a metric space is the minimum α such that any ball of radius r can be covered by at most 2 α balls of radius r/2.) However, all of these schemes are static and